IEEE Transactions on Automatic Control, Vol.53, No.10, 2266-2279, 2008
Frequency Analysis and Norms of Distributed Spatially Periodic Systems
We investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients. We develop a spatial-frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems. Using this representation, we introduce the notion of the H-2 norm for this class of systems and provide algorithms for its computation. A stochastic interpretation of the V norm is given in terms of spatially cyclostationary random fields and spectral-correlation density operators. When the periodic coefficients are viewed as feedback modifications of spatially invariant systems, we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems. Two examples from physics are provided to illustrate the main results.